# Chapter 4: Structural Equation Models

### Multigroup structural equation models

Like factor analysis models, SEMs can be generalized to a multigroup version to examine how parameters of the model may vary between groups such as countries. For the measurement models and the distribution of exogenous latent factors this is done as in multigroup factor analysis, in the ways that were explained in Chapter 3. So the only new element that needs to be explained here is the multigroup extension of structural regression models for endogenous factors. To illustrate this step, consider the model for factor η_{5} (willingness to co-operate with the police) in our example. The multigroup version of this model is

η_{5} = α_{5}^{(g)} + β_{51}^{(g)}η_{1} + β_{52}^{(g)}η_{2} + β_{53}^{(g)}η_{3} + β_{54}^{(g)}η_{4} + ζ_{5}

for a respondent in group *g* = 1, ..., *G* where *ζ*_{5} is normally distributed with mean 0 and variance *ψ*_{5}^{(g)}. In other words, all the parameters of this model may vary between the groups. The intercept and residual variance again need to be fixed in one group (for example, by fixing *α*_{5}^{(1)} = 0 and *ψ*_{5}^{(1)} = 1) but can be freely estimated in the other groups.

Likelihood ratio tests can be used to compare models where parameters do and do not vary between the groups, for example to test the model above against one where ψ_{5}^{(g)} = ψ_{5} in all groups *g*, (i.e. where the residual variance for *η*_{5} is the same across the groups).

Typically the most interesting parameters are the regression coefficients *β*. Cross-group variation in them indicates that associations between a response variable (e.g. *η*_{5} above) and its explanatory variables are of different strengths in different groups. In the language of regression models, such variation thus indicates an *interaction* between the group and the explanatory variables in a model. An illustration of the estimation and testing such interactions in a cross-national analysis is given in Example 2 of this chapter.